Properties

Label 1400.2.g.i.449.4
Level $1400$
Weight $2$
Character 1400.449
Analytic conductor $11.179$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1400 = 2^{3} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1400.g (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(11.1790562830\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{33})\)
Defining polynomial: \(x^{4} + 17 x^{2} + 64\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 280)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 449.4
Root \(3.37228i\) of defining polynomial
Character \(\chi\) \(=\) 1400.449
Dual form 1400.2.g.i.449.1

$q$-expansion

\(f(q)\) \(=\) \(q+3.37228i q^{3} -1.00000i q^{7} -8.37228 q^{9} +O(q^{10})\) \(q+3.37228i q^{3} -1.00000i q^{7} -8.37228 q^{9} +0.627719 q^{11} +1.37228i q^{13} +5.37228i q^{17} -6.74456 q^{19} +3.37228 q^{21} -6.74456i q^{23} -18.1168i q^{27} -1.37228 q^{29} -8.00000 q^{31} +2.11684i q^{33} -2.00000i q^{37} -4.62772 q^{39} -4.74456 q^{41} -2.74456i q^{43} +10.1168i q^{47} -1.00000 q^{49} -18.1168 q^{51} +0.744563i q^{53} -22.7446i q^{57} -8.00000 q^{59} +8.74456 q^{61} +8.37228i q^{63} -4.00000i q^{67} +22.7446 q^{69} +8.00000 q^{71} +6.00000i q^{73} -0.627719i q^{77} +2.11684 q^{79} +35.9783 q^{81} -13.4891i q^{83} -4.62772i q^{87} -3.25544 q^{89} +1.37228 q^{91} -26.9783i q^{93} +18.8614i q^{97} -5.25544 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 22q^{9} + O(q^{10}) \) \( 4q - 22q^{9} + 14q^{11} - 4q^{19} + 2q^{21} + 6q^{29} - 32q^{31} - 30q^{39} + 4q^{41} - 4q^{49} - 38q^{51} - 32q^{59} + 12q^{61} + 68q^{69} + 32q^{71} - 26q^{79} + 52q^{81} - 36q^{89} - 6q^{91} - 44q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1400\mathbb{Z}\right)^\times\).

\(n\) \(351\) \(701\) \(801\) \(1177\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 3.37228i 1.94699i 0.228714 + 0.973494i \(0.426548\pi\)
−0.228714 + 0.973494i \(0.573452\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) − 1.00000i − 0.377964i
\(8\) 0 0
\(9\) −8.37228 −2.79076
\(10\) 0 0
\(11\) 0.627719 0.189264 0.0946322 0.995512i \(-0.469833\pi\)
0.0946322 + 0.995512i \(0.469833\pi\)
\(12\) 0 0
\(13\) 1.37228i 0.380602i 0.981726 + 0.190301i \(0.0609465\pi\)
−0.981726 + 0.190301i \(0.939054\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 5.37228i 1.30297i 0.758662 + 0.651485i \(0.225854\pi\)
−0.758662 + 0.651485i \(0.774146\pi\)
\(18\) 0 0
\(19\) −6.74456 −1.54731 −0.773654 0.633608i \(-0.781573\pi\)
−0.773654 + 0.633608i \(0.781573\pi\)
\(20\) 0 0
\(21\) 3.37228 0.735892
\(22\) 0 0
\(23\) − 6.74456i − 1.40634i −0.711022 0.703169i \(-0.751768\pi\)
0.711022 0.703169i \(-0.248232\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) − 18.1168i − 3.48659i
\(28\) 0 0
\(29\) −1.37228 −0.254826 −0.127413 0.991850i \(-0.540667\pi\)
−0.127413 + 0.991850i \(0.540667\pi\)
\(30\) 0 0
\(31\) −8.00000 −1.43684 −0.718421 0.695608i \(-0.755135\pi\)
−0.718421 + 0.695608i \(0.755135\pi\)
\(32\) 0 0
\(33\) 2.11684i 0.368495i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) − 2.00000i − 0.328798i −0.986394 0.164399i \(-0.947432\pi\)
0.986394 0.164399i \(-0.0525685\pi\)
\(38\) 0 0
\(39\) −4.62772 −0.741028
\(40\) 0 0
\(41\) −4.74456 −0.740976 −0.370488 0.928837i \(-0.620810\pi\)
−0.370488 + 0.928837i \(0.620810\pi\)
\(42\) 0 0
\(43\) − 2.74456i − 0.418542i −0.977858 0.209271i \(-0.932891\pi\)
0.977858 0.209271i \(-0.0671091\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 10.1168i 1.47569i 0.674968 + 0.737847i \(0.264157\pi\)
−0.674968 + 0.737847i \(0.735843\pi\)
\(48\) 0 0
\(49\) −1.00000 −0.142857
\(50\) 0 0
\(51\) −18.1168 −2.53687
\(52\) 0 0
\(53\) 0.744563i 0.102274i 0.998692 + 0.0511368i \(0.0162844\pi\)
−0.998692 + 0.0511368i \(0.983716\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) − 22.7446i − 3.01259i
\(58\) 0 0
\(59\) −8.00000 −1.04151 −0.520756 0.853706i \(-0.674350\pi\)
−0.520756 + 0.853706i \(0.674350\pi\)
\(60\) 0 0
\(61\) 8.74456 1.11963 0.559813 0.828619i \(-0.310873\pi\)
0.559813 + 0.828619i \(0.310873\pi\)
\(62\) 0 0
\(63\) 8.37228i 1.05481i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) − 4.00000i − 0.488678i −0.969690 0.244339i \(-0.921429\pi\)
0.969690 0.244339i \(-0.0785709\pi\)
\(68\) 0 0
\(69\) 22.7446 2.73812
\(70\) 0 0
\(71\) 8.00000 0.949425 0.474713 0.880141i \(-0.342552\pi\)
0.474713 + 0.880141i \(0.342552\pi\)
\(72\) 0 0
\(73\) 6.00000i 0.702247i 0.936329 + 0.351123i \(0.114200\pi\)
−0.936329 + 0.351123i \(0.885800\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) − 0.627719i − 0.0715352i
\(78\) 0 0
\(79\) 2.11684 0.238164 0.119082 0.992884i \(-0.462005\pi\)
0.119082 + 0.992884i \(0.462005\pi\)
\(80\) 0 0
\(81\) 35.9783 3.99758
\(82\) 0 0
\(83\) − 13.4891i − 1.48062i −0.672264 0.740312i \(-0.734678\pi\)
0.672264 0.740312i \(-0.265322\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) − 4.62772i − 0.496144i
\(88\) 0 0
\(89\) −3.25544 −0.345076 −0.172538 0.985003i \(-0.555197\pi\)
−0.172538 + 0.985003i \(0.555197\pi\)
\(90\) 0 0
\(91\) 1.37228 0.143854
\(92\) 0 0
\(93\) − 26.9783i − 2.79751i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 18.8614i 1.91509i 0.288291 + 0.957543i \(0.406913\pi\)
−0.288291 + 0.957543i \(0.593087\pi\)
\(98\) 0 0
\(99\) −5.25544 −0.528191
\(100\) 0 0
\(101\) −6.00000 −0.597022 −0.298511 0.954406i \(-0.596490\pi\)
−0.298511 + 0.954406i \(0.596490\pi\)
\(102\) 0 0
\(103\) 11.3723i 1.12054i 0.828309 + 0.560272i \(0.189303\pi\)
−0.828309 + 0.560272i \(0.810697\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) − 2.74456i − 0.265327i −0.991161 0.132663i \(-0.957647\pi\)
0.991161 0.132663i \(-0.0423529\pi\)
\(108\) 0 0
\(109\) 5.37228 0.514571 0.257286 0.966335i \(-0.417172\pi\)
0.257286 + 0.966335i \(0.417172\pi\)
\(110\) 0 0
\(111\) 6.74456 0.640166
\(112\) 0 0
\(113\) − 2.00000i − 0.188144i −0.995565 0.0940721i \(-0.970012\pi\)
0.995565 0.0940721i \(-0.0299884\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) − 11.4891i − 1.06217i
\(118\) 0 0
\(119\) 5.37228 0.492476
\(120\) 0 0
\(121\) −10.6060 −0.964179
\(122\) 0 0
\(123\) − 16.0000i − 1.44267i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 8.00000i 0.709885i 0.934888 + 0.354943i \(0.115500\pi\)
−0.934888 + 0.354943i \(0.884500\pi\)
\(128\) 0 0
\(129\) 9.25544 0.814896
\(130\) 0 0
\(131\) 6.74456 0.589275 0.294638 0.955609i \(-0.404801\pi\)
0.294638 + 0.955609i \(0.404801\pi\)
\(132\) 0 0
\(133\) 6.74456i 0.584828i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 3.25544i 0.278131i 0.990283 + 0.139065i \(0.0444098\pi\)
−0.990283 + 0.139065i \(0.955590\pi\)
\(138\) 0 0
\(139\) −6.74456 −0.572066 −0.286033 0.958220i \(-0.592337\pi\)
−0.286033 + 0.958220i \(0.592337\pi\)
\(140\) 0 0
\(141\) −34.1168 −2.87316
\(142\) 0 0
\(143\) 0.861407i 0.0720344i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) − 3.37228i − 0.278141i
\(148\) 0 0
\(149\) 7.48913 0.613533 0.306767 0.951785i \(-0.400753\pi\)
0.306767 + 0.951785i \(0.400753\pi\)
\(150\) 0 0
\(151\) −2.11684 −0.172266 −0.0861332 0.996284i \(-0.527451\pi\)
−0.0861332 + 0.996284i \(0.527451\pi\)
\(152\) 0 0
\(153\) − 44.9783i − 3.63628i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 7.48913i 0.597697i 0.954301 + 0.298849i \(0.0966026\pi\)
−0.954301 + 0.298849i \(0.903397\pi\)
\(158\) 0 0
\(159\) −2.51087 −0.199125
\(160\) 0 0
\(161\) −6.74456 −0.531546
\(162\) 0 0
\(163\) 5.25544i 0.411638i 0.978590 + 0.205819i \(0.0659858\pi\)
−0.978590 + 0.205819i \(0.934014\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) − 11.3723i − 0.880014i −0.897994 0.440007i \(-0.854976\pi\)
0.897994 0.440007i \(-0.145024\pi\)
\(168\) 0 0
\(169\) 11.1168 0.855142
\(170\) 0 0
\(171\) 56.4674 4.31817
\(172\) 0 0
\(173\) − 5.37228i − 0.408447i −0.978924 0.204223i \(-0.934533\pi\)
0.978924 0.204223i \(-0.0654669\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) − 26.9783i − 2.02781i
\(178\) 0 0
\(179\) −22.9783 −1.71748 −0.858738 0.512416i \(-0.828751\pi\)
−0.858738 + 0.512416i \(0.828751\pi\)
\(180\) 0 0
\(181\) −18.2337 −1.35530 −0.677650 0.735385i \(-0.737001\pi\)
−0.677650 + 0.735385i \(0.737001\pi\)
\(182\) 0 0
\(183\) 29.4891i 2.17990i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 3.37228i 0.246606i
\(188\) 0 0
\(189\) −18.1168 −1.31781
\(190\) 0 0
\(191\) −24.8614 −1.79891 −0.899454 0.437015i \(-0.856036\pi\)
−0.899454 + 0.437015i \(0.856036\pi\)
\(192\) 0 0
\(193\) 4.74456i 0.341521i 0.985313 + 0.170761i \(0.0546225\pi\)
−0.985313 + 0.170761i \(0.945378\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 26.2337i 1.86907i 0.355867 + 0.934536i \(0.384185\pi\)
−0.355867 + 0.934536i \(0.615815\pi\)
\(198\) 0 0
\(199\) 16.0000 1.13421 0.567105 0.823646i \(-0.308063\pi\)
0.567105 + 0.823646i \(0.308063\pi\)
\(200\) 0 0
\(201\) 13.4891 0.951450
\(202\) 0 0
\(203\) 1.37228i 0.0963153i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 56.4674i 3.92475i
\(208\) 0 0
\(209\) −4.23369 −0.292850
\(210\) 0 0
\(211\) −8.62772 −0.593957 −0.296978 0.954884i \(-0.595979\pi\)
−0.296978 + 0.954884i \(0.595979\pi\)
\(212\) 0 0
\(213\) 26.9783i 1.84852i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 8.00000i 0.543075i
\(218\) 0 0
\(219\) −20.2337 −1.36727
\(220\) 0 0
\(221\) −7.37228 −0.495913
\(222\) 0 0
\(223\) 11.3723i 0.761544i 0.924669 + 0.380772i \(0.124342\pi\)
−0.924669 + 0.380772i \(0.875658\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) − 8.86141i − 0.588152i −0.955782 0.294076i \(-0.904988\pi\)
0.955782 0.294076i \(-0.0950119\pi\)
\(228\) 0 0
\(229\) −22.2337 −1.46924 −0.734622 0.678477i \(-0.762640\pi\)
−0.734622 + 0.678477i \(0.762640\pi\)
\(230\) 0 0
\(231\) 2.11684 0.139278
\(232\) 0 0
\(233\) 12.7446i 0.834924i 0.908694 + 0.417462i \(0.137080\pi\)
−0.908694 + 0.417462i \(0.862920\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 7.13859i 0.463701i
\(238\) 0 0
\(239\) −19.3723 −1.25309 −0.626544 0.779386i \(-0.715531\pi\)
−0.626544 + 0.779386i \(0.715531\pi\)
\(240\) 0 0
\(241\) 26.0000 1.67481 0.837404 0.546585i \(-0.184072\pi\)
0.837404 + 0.546585i \(0.184072\pi\)
\(242\) 0 0
\(243\) 66.9783i 4.29666i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) − 9.25544i − 0.588909i
\(248\) 0 0
\(249\) 45.4891 2.88276
\(250\) 0 0
\(251\) −6.74456 −0.425713 −0.212857 0.977083i \(-0.568277\pi\)
−0.212857 + 0.977083i \(0.568277\pi\)
\(252\) 0 0
\(253\) − 4.23369i − 0.266170i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) − 0.510875i − 0.0318675i −0.999873 0.0159337i \(-0.994928\pi\)
0.999873 0.0159337i \(-0.00507208\pi\)
\(258\) 0 0
\(259\) −2.00000 −0.124274
\(260\) 0 0
\(261\) 11.4891 0.711159
\(262\) 0 0
\(263\) 12.2337i 0.754362i 0.926140 + 0.377181i \(0.123106\pi\)
−0.926140 + 0.377181i \(0.876894\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) − 10.9783i − 0.671858i
\(268\) 0 0
\(269\) 6.00000 0.365826 0.182913 0.983129i \(-0.441447\pi\)
0.182913 + 0.983129i \(0.441447\pi\)
\(270\) 0 0
\(271\) −13.4891 −0.819406 −0.409703 0.912219i \(-0.634368\pi\)
−0.409703 + 0.912219i \(0.634368\pi\)
\(272\) 0 0
\(273\) 4.62772i 0.280082i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) − 20.9783i − 1.26046i −0.776408 0.630230i \(-0.782960\pi\)
0.776408 0.630230i \(-0.217040\pi\)
\(278\) 0 0
\(279\) 66.9783 4.00988
\(280\) 0 0
\(281\) −21.6060 −1.28890 −0.644452 0.764645i \(-0.722914\pi\)
−0.644452 + 0.764645i \(0.722914\pi\)
\(282\) 0 0
\(283\) 26.1168i 1.55249i 0.630434 + 0.776243i \(0.282877\pi\)
−0.630434 + 0.776243i \(0.717123\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 4.74456i 0.280063i
\(288\) 0 0
\(289\) −11.8614 −0.697730
\(290\) 0 0
\(291\) −63.6060 −3.72865
\(292\) 0 0
\(293\) − 7.88316i − 0.460539i −0.973127 0.230269i \(-0.926039\pi\)
0.973127 0.230269i \(-0.0739608\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) − 11.3723i − 0.659887i
\(298\) 0 0
\(299\) 9.25544 0.535256
\(300\) 0 0
\(301\) −2.74456 −0.158194
\(302\) 0 0
\(303\) − 20.2337i − 1.16240i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 13.8832i 0.792354i 0.918174 + 0.396177i \(0.129663\pi\)
−0.918174 + 0.396177i \(0.870337\pi\)
\(308\) 0 0
\(309\) −38.3505 −2.18169
\(310\) 0 0
\(311\) −1.25544 −0.0711893 −0.0355947 0.999366i \(-0.511333\pi\)
−0.0355947 + 0.999366i \(0.511333\pi\)
\(312\) 0 0
\(313\) − 20.1168i − 1.13707i −0.822659 0.568536i \(-0.807510\pi\)
0.822659 0.568536i \(-0.192490\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 14.0000i 0.786318i 0.919470 + 0.393159i \(0.128618\pi\)
−0.919470 + 0.393159i \(0.871382\pi\)
\(318\) 0 0
\(319\) −0.861407 −0.0482295
\(320\) 0 0
\(321\) 9.25544 0.516588
\(322\) 0 0
\(323\) − 36.2337i − 2.01610i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 18.1168i 1.00186i
\(328\) 0 0
\(329\) 10.1168 0.557760
\(330\) 0 0
\(331\) 12.0000 0.659580 0.329790 0.944054i \(-0.393022\pi\)
0.329790 + 0.944054i \(0.393022\pi\)
\(332\) 0 0
\(333\) 16.7446i 0.917596i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 15.4891i 0.843746i 0.906655 + 0.421873i \(0.138627\pi\)
−0.906655 + 0.421873i \(0.861373\pi\)
\(338\) 0 0
\(339\) 6.74456 0.366314
\(340\) 0 0
\(341\) −5.02175 −0.271943
\(342\) 0 0
\(343\) 1.00000i 0.0539949i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) − 13.2554i − 0.711589i −0.934564 0.355795i \(-0.884210\pi\)
0.934564 0.355795i \(-0.115790\pi\)
\(348\) 0 0
\(349\) 3.48913 0.186769 0.0933843 0.995630i \(-0.470231\pi\)
0.0933843 + 0.995630i \(0.470231\pi\)
\(350\) 0 0
\(351\) 24.8614 1.32700
\(352\) 0 0
\(353\) − 26.8614i − 1.42969i −0.699284 0.714844i \(-0.746498\pi\)
0.699284 0.714844i \(-0.253502\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 18.1168i 0.958845i
\(358\) 0 0
\(359\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(360\) 0 0
\(361\) 26.4891 1.39416
\(362\) 0 0
\(363\) − 35.7663i − 1.87724i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) − 8.86141i − 0.462562i −0.972887 0.231281i \(-0.925708\pi\)
0.972887 0.231281i \(-0.0742916\pi\)
\(368\) 0 0
\(369\) 39.7228 2.06789
\(370\) 0 0
\(371\) 0.744563 0.0386558
\(372\) 0 0
\(373\) 19.2554i 0.997009i 0.866887 + 0.498504i \(0.166117\pi\)
−0.866887 + 0.498504i \(0.833883\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) − 1.88316i − 0.0969875i
\(378\) 0 0
\(379\) −12.0000 −0.616399 −0.308199 0.951322i \(-0.599726\pi\)
−0.308199 + 0.951322i \(0.599726\pi\)
\(380\) 0 0
\(381\) −26.9783 −1.38214
\(382\) 0 0
\(383\) − 5.48913i − 0.280481i −0.990117 0.140241i \(-0.955212\pi\)
0.990117 0.140241i \(-0.0447876\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 22.9783i 1.16805i
\(388\) 0 0
\(389\) 10.8614 0.550695 0.275348 0.961345i \(-0.411207\pi\)
0.275348 + 0.961345i \(0.411207\pi\)
\(390\) 0 0
\(391\) 36.2337 1.83242
\(392\) 0 0
\(393\) 22.7446i 1.14731i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 37.3723i 1.87566i 0.347094 + 0.937831i \(0.387169\pi\)
−0.347094 + 0.937831i \(0.612831\pi\)
\(398\) 0 0
\(399\) −22.7446 −1.13865
\(400\) 0 0
\(401\) 1.60597 0.0801983 0.0400991 0.999196i \(-0.487233\pi\)
0.0400991 + 0.999196i \(0.487233\pi\)
\(402\) 0 0
\(403\) − 10.9783i − 0.546866i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) − 1.25544i − 0.0622297i
\(408\) 0 0
\(409\) 11.4891 0.568101 0.284050 0.958809i \(-0.408322\pi\)
0.284050 + 0.958809i \(0.408322\pi\)
\(410\) 0 0
\(411\) −10.9783 −0.541517
\(412\) 0 0
\(413\) 8.00000i 0.393654i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) − 22.7446i − 1.11381i
\(418\) 0 0
\(419\) 37.4891 1.83146 0.915732 0.401790i \(-0.131612\pi\)
0.915732 + 0.401790i \(0.131612\pi\)
\(420\) 0 0
\(421\) 21.6060 1.05301 0.526505 0.850172i \(-0.323502\pi\)
0.526505 + 0.850172i \(0.323502\pi\)
\(422\) 0 0
\(423\) − 84.7011i − 4.11831i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) − 8.74456i − 0.423179i
\(428\) 0 0
\(429\) −2.90491 −0.140250
\(430\) 0 0
\(431\) 12.6277 0.608256 0.304128 0.952631i \(-0.401635\pi\)
0.304128 + 0.952631i \(0.401635\pi\)
\(432\) 0 0
\(433\) 16.9783i 0.815923i 0.912999 + 0.407961i \(0.133760\pi\)
−0.912999 + 0.407961i \(0.866240\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 45.4891i 2.17604i
\(438\) 0 0
\(439\) −28.2337 −1.34752 −0.673760 0.738950i \(-0.735322\pi\)
−0.673760 + 0.738950i \(0.735322\pi\)
\(440\) 0 0
\(441\) 8.37228 0.398680
\(442\) 0 0
\(443\) 5.25544i 0.249693i 0.992176 + 0.124847i \(0.0398439\pi\)
−0.992176 + 0.124847i \(0.960156\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 25.2554i 1.19454i
\(448\) 0 0
\(449\) 0.116844 0.00551421 0.00275710 0.999996i \(-0.499122\pi\)
0.00275710 + 0.999996i \(0.499122\pi\)
\(450\) 0 0
\(451\) −2.97825 −0.140240
\(452\) 0 0
\(453\) − 7.13859i − 0.335400i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 16.7446i 0.783278i 0.920119 + 0.391639i \(0.128092\pi\)
−0.920119 + 0.391639i \(0.871908\pi\)
\(458\) 0 0
\(459\) 97.3288 4.54292
\(460\) 0 0
\(461\) −12.7446 −0.593573 −0.296787 0.954944i \(-0.595915\pi\)
−0.296787 + 0.954944i \(0.595915\pi\)
\(462\) 0 0
\(463\) 29.4891i 1.37048i 0.728319 + 0.685238i \(0.240302\pi\)
−0.728319 + 0.685238i \(0.759698\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) − 31.6060i − 1.46255i −0.682083 0.731275i \(-0.738926\pi\)
0.682083 0.731275i \(-0.261074\pi\)
\(468\) 0 0
\(469\) −4.00000 −0.184703
\(470\) 0 0
\(471\) −25.2554 −1.16371
\(472\) 0 0
\(473\) − 1.72281i − 0.0792150i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) − 6.23369i − 0.285421i
\(478\) 0 0
\(479\) 12.2337 0.558971 0.279486 0.960150i \(-0.409836\pi\)
0.279486 + 0.960150i \(0.409836\pi\)
\(480\) 0 0
\(481\) 2.74456 0.125141
\(482\) 0 0
\(483\) − 22.7446i − 1.03491i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) − 4.23369i − 0.191847i −0.995389 0.0959234i \(-0.969420\pi\)
0.995389 0.0959234i \(-0.0305804\pi\)
\(488\) 0 0
\(489\) −17.7228 −0.801453
\(490\) 0 0
\(491\) 17.8832 0.807056 0.403528 0.914967i \(-0.367784\pi\)
0.403528 + 0.914967i \(0.367784\pi\)
\(492\) 0 0
\(493\) − 7.37228i − 0.332031i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) − 8.00000i − 0.358849i
\(498\) 0 0
\(499\) −3.13859 −0.140503 −0.0702514 0.997529i \(-0.522380\pi\)
−0.0702514 + 0.997529i \(0.522380\pi\)
\(500\) 0 0
\(501\) 38.3505 1.71338
\(502\) 0 0
\(503\) 12.6277i 0.563042i 0.959555 + 0.281521i \(0.0908389\pi\)
−0.959555 + 0.281521i \(0.909161\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 37.4891i 1.66495i
\(508\) 0 0
\(509\) −4.97825 −0.220657 −0.110329 0.993895i \(-0.535190\pi\)
−0.110329 + 0.993895i \(0.535190\pi\)
\(510\) 0 0
\(511\) 6.00000 0.265424
\(512\) 0 0
\(513\) 122.190i 5.39483i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 6.35053i 0.279296i
\(518\) 0 0
\(519\) 18.1168 0.795241
\(520\) 0 0
\(521\) −30.0000 −1.31432 −0.657162 0.753749i \(-0.728243\pi\)
−0.657162 + 0.753749i \(0.728243\pi\)
\(522\) 0 0
\(523\) − 32.4674i − 1.41970i −0.704353 0.709850i \(-0.748763\pi\)
0.704353 0.709850i \(-0.251237\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) − 42.9783i − 1.87216i
\(528\) 0 0
\(529\) −22.4891 −0.977788
\(530\) 0 0
\(531\) 66.9783 2.90661
\(532\) 0 0
\(533\) − 6.51087i − 0.282017i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) − 77.4891i − 3.34390i
\(538\) 0 0
\(539\) −0.627719 −0.0270378
\(540\) 0 0
\(541\) 20.3505 0.874938 0.437469 0.899234i \(-0.355875\pi\)
0.437469 + 0.899234i \(0.355875\pi\)
\(542\) 0 0
\(543\) − 61.4891i − 2.63875i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 14.9783i 0.640424i 0.947346 + 0.320212i \(0.103754\pi\)
−0.947346 + 0.320212i \(0.896246\pi\)
\(548\) 0 0
\(549\) −73.2119 −3.12461
\(550\) 0 0
\(551\) 9.25544 0.394295
\(552\) 0 0
\(553\) − 2.11684i − 0.0900174i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) − 38.2337i − 1.62001i −0.586421 0.810007i \(-0.699463\pi\)
0.586421 0.810007i \(-0.300537\pi\)
\(558\) 0 0
\(559\) 3.76631 0.159298
\(560\) 0 0
\(561\) −11.3723 −0.480138
\(562\) 0 0
\(563\) 5.48913i 0.231339i 0.993288 + 0.115670i \(0.0369014\pi\)
−0.993288 + 0.115670i \(0.963099\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) − 35.9783i − 1.51094i
\(568\) 0 0
\(569\) −20.9783 −0.879454 −0.439727 0.898131i \(-0.644925\pi\)
−0.439727 + 0.898131i \(0.644925\pi\)
\(570\) 0 0
\(571\) −20.0000 −0.836974 −0.418487 0.908223i \(-0.637439\pi\)
−0.418487 + 0.908223i \(0.637439\pi\)
\(572\) 0 0
\(573\) − 83.8397i − 3.50245i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 17.6060i 0.732946i 0.930429 + 0.366473i \(0.119435\pi\)
−0.930429 + 0.366473i \(0.880565\pi\)
\(578\) 0 0
\(579\) −16.0000 −0.664937
\(580\) 0 0
\(581\) −13.4891 −0.559623
\(582\) 0 0
\(583\) 0.467376i 0.0193567i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 10.9783i 0.453121i 0.973997 + 0.226560i \(0.0727481\pi\)
−0.973997 + 0.226560i \(0.927252\pi\)
\(588\) 0 0
\(589\) 53.9565 2.22324
\(590\) 0 0
\(591\) −88.4674 −3.63906
\(592\) 0 0
\(593\) 25.3723i 1.04191i 0.853583 + 0.520957i \(0.174425\pi\)
−0.853583 + 0.520957i \(0.825575\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 53.9565i 2.20829i
\(598\) 0 0
\(599\) −7.60597 −0.310771 −0.155386 0.987854i \(-0.549662\pi\)
−0.155386 + 0.987854i \(0.549662\pi\)
\(600\) 0 0
\(601\) 39.4891 1.61080 0.805398 0.592735i \(-0.201952\pi\)
0.805398 + 0.592735i \(0.201952\pi\)
\(602\) 0 0
\(603\) 33.4891i 1.36378i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) − 15.6060i − 0.633427i −0.948521 0.316713i \(-0.897421\pi\)
0.948521 0.316713i \(-0.102579\pi\)
\(608\) 0 0
\(609\) −4.62772 −0.187525
\(610\) 0 0
\(611\) −13.8832 −0.561652
\(612\) 0 0
\(613\) 31.4891i 1.27183i 0.771758 + 0.635917i \(0.219378\pi\)
−0.771758 + 0.635917i \(0.780622\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) − 6.00000i − 0.241551i −0.992680 0.120775i \(-0.961462\pi\)
0.992680 0.120775i \(-0.0385381\pi\)
\(618\) 0 0
\(619\) 1.25544 0.0504603 0.0252301 0.999682i \(-0.491968\pi\)
0.0252301 + 0.999682i \(0.491968\pi\)
\(620\) 0 0
\(621\) −122.190 −4.90332
\(622\) 0 0
\(623\) 3.25544i 0.130426i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) − 14.2772i − 0.570176i
\(628\) 0 0
\(629\) 10.7446 0.428414
\(630\) 0 0
\(631\) −3.37228 −0.134248 −0.0671242 0.997745i \(-0.521382\pi\)
−0.0671242 + 0.997745i \(0.521382\pi\)
\(632\) 0 0
\(633\) − 29.0951i − 1.15643i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) − 1.37228i − 0.0543718i
\(638\) 0 0
\(639\) −66.9783 −2.64962
\(640\) 0 0
\(641\) 2.00000 0.0789953 0.0394976 0.999220i \(-0.487424\pi\)
0.0394976 + 0.999220i \(0.487424\pi\)
\(642\) 0 0
\(643\) − 12.6277i − 0.497989i −0.968505 0.248994i \(-0.919900\pi\)
0.968505 0.248994i \(-0.0801000\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) − 16.0000i − 0.629025i −0.949253 0.314512i \(-0.898159\pi\)
0.949253 0.314512i \(-0.101841\pi\)
\(648\) 0 0
\(649\) −5.02175 −0.197121
\(650\) 0 0
\(651\) −26.9783 −1.05736
\(652\) 0 0
\(653\) − 6.00000i − 0.234798i −0.993085 0.117399i \(-0.962544\pi\)
0.993085 0.117399i \(-0.0374557\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) − 50.2337i − 1.95980i
\(658\) 0 0
\(659\) −6.11684 −0.238278 −0.119139 0.992878i \(-0.538013\pi\)
−0.119139 + 0.992878i \(0.538013\pi\)
\(660\) 0 0
\(661\) 3.25544 0.126622 0.0633109 0.997994i \(-0.479834\pi\)
0.0633109 + 0.997994i \(0.479834\pi\)
\(662\) 0 0
\(663\) − 24.8614i − 0.965537i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 9.25544i 0.358372i
\(668\) 0 0
\(669\) −38.3505 −1.48272
\(670\) 0 0
\(671\) 5.48913 0.211905
\(672\) 0 0
\(673\) 31.7228i 1.22282i 0.791312 + 0.611412i \(0.209398\pi\)
−0.791312 + 0.611412i \(0.790602\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) − 36.3505i − 1.39706i −0.715579 0.698532i \(-0.753837\pi\)
0.715579 0.698532i \(-0.246163\pi\)
\(678\) 0 0
\(679\) 18.8614 0.723834
\(680\) 0 0
\(681\) 29.8832 1.14513
\(682\) 0 0
\(683\) − 36.0000i − 1.37750i −0.724998 0.688751i \(-0.758159\pi\)
0.724998 0.688751i \(-0.241841\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) − 74.9783i − 2.86060i
\(688\) 0 0
\(689\) −1.02175 −0.0389256
\(690\) 0 0
\(691\) 13.4891 0.513151 0.256575 0.966524i \(-0.417406\pi\)
0.256575 + 0.966524i \(0.417406\pi\)
\(692\) 0 0
\(693\) 5.25544i 0.199638i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) − 25.4891i − 0.965469i
\(698\) 0 0
\(699\) −42.9783 −1.62559
\(700\) 0 0
\(701\) 30.8614 1.16562 0.582810 0.812609i \(-0.301953\pi\)
0.582810 + 0.812609i \(0.301953\pi\)
\(702\) 0 0
\(703\) 13.4891i 0.508752i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 6.00000i 0.225653i
\(708\) 0 0
\(709\) 33.6060 1.26210 0.631049 0.775743i \(-0.282625\pi\)
0.631049 + 0.775743i \(0.282625\pi\)
\(710\) 0 0
\(711\) −17.7228 −0.664657
\(712\) 0 0
\(713\) 53.9565i 2.02069i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) − 65.3288i − 2.43975i
\(718\) 0 0
\(719\) −14.7446 −0.549879 −0.274940 0.961461i \(-0.588658\pi\)
−0.274940 + 0.961461i \(0.588658\pi\)
\(720\) 0 0
\(721\) 11.3723 0.423526
\(722\) 0 0
\(723\) 87.6793i 3.26083i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 32.0000i 1.18681i 0.804902 + 0.593407i \(0.202218\pi\)
−0.804902 + 0.593407i \(0.797782\pi\)
\(728\) 0 0
\(729\) −117.935 −4.36795
\(730\) 0 0
\(731\) 14.7446 0.545347
\(732\) 0 0
\(733\) 38.8614i 1.43538i 0.696363 + 0.717689i \(0.254800\pi\)
−0.696363 + 0.717689i \(0.745200\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) − 2.51087i − 0.0924893i
\(738\) 0 0
\(739\) −19.6060 −0.721217 −0.360609 0.932717i \(-0.617431\pi\)
−0.360609 + 0.932717i \(0.617431\pi\)
\(740\) 0 0
\(741\) 31.2119 1.14660
\(742\) 0 0
\(743\) 29.4891i 1.08185i 0.841070 + 0.540926i \(0.181926\pi\)
−0.841070 + 0.540926i \(0.818074\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 112.935i 4.13207i
\(748\) 0 0
\(749\) −2.74456 −0.100284
\(750\) 0 0
\(751\) 48.8614 1.78298 0.891489 0.453042i \(-0.149661\pi\)
0.891489 + 0.453042i \(0.149661\pi\)
\(752\) 0 0
\(753\) − 22.7446i − 0.828858i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) − 38.2337i − 1.38963i −0.719190 0.694814i \(-0.755487\pi\)
0.719190 0.694814i \(-0.244513\pi\)
\(758\) 0 0
\(759\) 14.2772 0.518229
\(760\) 0 0
\(761\) −40.9783 −1.48546 −0.742730 0.669591i \(-0.766469\pi\)
−0.742730 + 0.669591i \(0.766469\pi\)
\(762\) 0 0
\(763\) − 5.37228i − 0.194490i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) − 10.9783i − 0.396402i
\(768\) 0 0
\(769\) 19.4891 0.702796 0.351398 0.936226i \(-0.385706\pi\)
0.351398 + 0.936226i \(0.385706\pi\)
\(770\) 0 0
\(771\) 1.72281 0.0620456
\(772\) 0 0
\(773\) 1.37228i 0.0493575i 0.999695 + 0.0246788i \(0.00785629\pi\)
−0.999695 + 0.0246788i \(0.992144\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) − 6.74456i − 0.241960i
\(778\) 0 0
\(779\) 32.0000 1.14652
\(780\) 0 0
\(781\) 5.02175 0.179692
\(782\) 0 0
\(783\) 24.8614i 0.888474i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) − 13.8832i − 0.494881i −0.968903 0.247441i \(-0.920411\pi\)
0.968903 0.247441i \(-0.0795895\pi\)
\(788\) 0 0
\(789\) −41.2554 −1.46873
\(790\) 0 0
\(791\) −2.00000 −0.0711118
\(792\) 0 0
\(793\) 12.0000i 0.426132i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 21.3723i 0.757045i 0.925592 + 0.378523i \(0.123568\pi\)
−0.925592 + 0.378523i \(0.876432\pi\)
\(798\) 0 0
\(799\) −54.3505 −1.92278
\(800\) 0 0
\(801\) 27.2554 0.963024
\(802\) 0 0
\(803\) 3.76631i 0.132910i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 20.2337i 0.712260i
\(808\) 0 0
\(809\) −14.6277 −0.514283 −0.257142 0.966374i \(-0.582781\pi\)
−0.257142 + 0.966374i \(0.582781\pi\)
\(810\) 0 0
\(811\) 1.25544 0.0440844 0.0220422 0.999757i \(-0.492983\pi\)
0.0220422 + 0.999757i \(0.492983\pi\)
\(812\) 0 0
\(813\) − 45.4891i − 1.59537i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 18.5109i 0.647614i
\(818\) 0 0
\(819\) −11.4891 −0.401463
\(820\) 0 0
\(821\) −7.88316 −0.275124 −0.137562 0.990493i \(-0.543927\pi\)
−0.137562 + 0.990493i \(0.543927\pi\)
\(822\) 0 0
\(823\) 8.00000i 0.278862i 0.990232 + 0.139431i \(0.0445274\pi\)
−0.990232 + 0.139431i \(0.955473\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 13.2554i 0.460937i 0.973080 + 0.230468i \(0.0740258\pi\)
−0.973080 + 0.230468i \(0.925974\pi\)
\(828\) 0 0
\(829\) −22.2337 −0.772208 −0.386104 0.922455i \(-0.626179\pi\)
−0.386104 + 0.922455i \(0.626179\pi\)
\(830\) 0 0
\(831\) 70.7446 2.45410
\(832\) 0 0
\(833\) − 5.37228i − 0.186139i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 144.935i 5.00968i
\(838\) 0 0
\(839\) 22.7446 0.785230 0.392615 0.919703i \(-0.371571\pi\)
0.392615 + 0.919703i \(0.371571\pi\)
\(840\) 0 0
\(841\) −27.1168 −0.935064
\(842\) 0 0
\(843\) − 72.8614i − 2.50948i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 10.6060i 0.364425i
\(848\) 0 0
\(849\) −88.0733 −3.02267
\(850\) 0 0
\(851\) −13.4891 −0.462401
\(852\) 0 0
\(853\) 16.5109i 0.565322i 0.959220 + 0.282661i \(0.0912171\pi\)
−0.959220 + 0.282661i \(0.908783\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 18.0000i 0.614868i 0.951569 + 0.307434i \(0.0994704\pi\)
−0.951569 + 0.307434i \(0.900530\pi\)
\(858\) 0 0
\(859\) 24.4674 0.834816 0.417408 0.908719i \(-0.362939\pi\)
0.417408 + 0.908719i \(0.362939\pi\)
\(860\) 0 0
\(861\) −16.0000 −0.545279
\(862\) 0 0
\(863\) − 2.51087i − 0.0854712i −0.999086 0.0427356i \(-0.986393\pi\)
0.999086 0.0427356i \(-0.0136073\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) − 40.0000i − 1.35847i
\(868\) 0 0
\(869\) 1.32878 0.0450759
\(870\) 0 0
\(871\) 5.48913 0.185992
\(872\) 0 0
\(873\) − 157.913i − 5.34455i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) − 52.9783i − 1.78895i −0.447120 0.894474i \(-0.647550\pi\)
0.447120 0.894474i \(-0.352450\pi\)
\(878\) 0 0
\(879\) 26.5842 0.896663
\(880\) 0 0
\(881\) −36.7446 −1.23796 −0.618978 0.785408i \(-0.712453\pi\)
−0.618978 + 0.785408i \(0.712453\pi\)
\(882\) 0 0
\(883\) 33.4891i 1.12700i 0.826116 + 0.563499i \(0.190545\pi\)
−0.826116 + 0.563499i \(0.809455\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) − 58.9783i − 1.98030i −0.140025 0.990148i \(-0.544718\pi\)
0.140025 0.990148i \(-0.455282\pi\)
\(888\) 0 0
\(889\) 8.00000 0.268311
\(890\) 0 0
\(891\) 22.5842 0.756600
\(892\) 0 0
\(893\) − 68.2337i − 2.28335i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 31.2119i 1.04214i
\(898\) 0 0
\(899\) 10.9783 0.366145
\(900\) 0 0
\(901\) −4.00000 −0.133259
\(902\) 0 0
\(903\) − 9.25544i − 0.308002i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) − 45.7228i − 1.51820i −0.650973 0.759101i \(-0.725639\pi\)
0.650973 0.759101i \(-0.274361\pi\)
\(908\) 0 0
\(909\) 50.2337 1.66615
\(910\) 0 0
\(911\) 45.9565 1.52261 0.761303 0.648396i \(-0.224560\pi\)
0.761303 + 0.648396i \(0.224560\pi\)
\(912\) 0 0
\(913\) − 8.46738i − 0.280229i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) − 6.74456i − 0.222725i
\(918\) 0 0
\(919\) −19.3723 −0.639033 −0.319516 0.947581i \(-0.603520\pi\)
−0.319516 + 0.947581i \(0.603520\pi\)
\(920\) 0 0
\(921\) −46.8179 −1.54270
\(922\) 0 0
\(923\) 10.9783i 0.361354i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) − 95.2119i − 3.12717i
\(928\) 0 0
\(929\) 42.2337 1.38564 0.692821 0.721109i \(-0.256368\pi\)
0.692821 + 0.721109i \(0.256368\pi\)
\(930\) 0 0
\(931\) 6.74456 0.221044
\(932\) 0 0
\(933\) − 4.23369i − 0.138605i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) − 35.0951i − 1.14651i −0.819378 0.573253i \(-0.805681\pi\)
0.819378 0.573253i \(-0.194319\pi\)
\(938\) 0 0
\(939\) 67.8397 2.21386
\(940\) 0 0
\(941\) −2.23369 −0.0728161 −0.0364081 0.999337i \(-0.511592\pi\)
−0.0364081 + 0.999337i \(0.511592\pi\)
\(942\) 0 0
\(943\) 32.0000i 1.04206i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) − 28.0000i − 0.909878i −0.890523 0.454939i \(-0.849661\pi\)
0.890523 0.454939i \(-0.150339\pi\)
\(948\) 0 0
\(949\) −8.23369 −0.267277
\(950\) 0 0
\(951\) −47.2119 −1.53095
\(952\) 0 0
\(953\) − 48.7446i − 1.57899i −0.613756 0.789496i \(-0.710342\pi\)
0.613756 0.789496i \(-0.289658\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) − 2.90491i − 0.0939023i
\(958\) 0 0
\(959\) 3.25544 0.105124
\(960\) 0 0
\(961\) 33.0000 1.06452
\(962\) 0 0
\(963\) 22.9783i 0.740464i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 36.2337i 1.16520i 0.812760 + 0.582598i \(0.197964\pi\)
−0.812760 + 0.582598i \(0.802036\pi\)
\(968\) 0 0
\(969\) 122.190 3.92531
\(970\) 0 0
\(971\) 10.5109 0.337310 0.168655 0.985675i \(-0.446058\pi\)
0.168655 + 0.985675i \(0.446058\pi\)
\(972\) 0 0
\(973\) 6.74456i 0.216221i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 7.02175i 0.224646i 0.993672 + 0.112323i \(0.0358291\pi\)
−0.993672 + 0.112323i \(0.964171\pi\)
\(978\) 0 0
\(979\) −2.04350 −0.0653105
\(980\) 0 0
\(981\) −44.9783 −1.43605
\(982\) 0 0
\(983\) 4.62772i 0.147601i 0.997273 + 0.0738007i \(0.0235129\pi\)
−0.997273 + 0.0738007i \(0.976487\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 34.1168i 1.08595i
\(988\) 0 0
\(989\) −18.5109 −0.588612
\(990\) 0 0
\(991\) 53.9565 1.71398 0.856992 0.515329i \(-0.172330\pi\)
0.856992 + 0.515329i \(0.172330\pi\)
\(992\) 0 0
\(993\) 40.4674i 1.28419i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) − 24.1168i − 0.763788i −0.924206 0.381894i \(-0.875272\pi\)
0.924206 0.381894i \(-0.124728\pi\)
\(998\) 0 0
\(999\) −36.2337 −1.14638
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 1400.2.g.i.449.4 4
4.3 odd 2 2800.2.g.r.449.1 4
5.2 odd 4 1400.2.a.r.1.2 2
5.3 odd 4 280.2.a.c.1.1 2
5.4 even 2 inner 1400.2.g.i.449.1 4
15.8 even 4 2520.2.a.x.1.2 2
20.3 even 4 560.2.a.h.1.2 2
20.7 even 4 2800.2.a.bk.1.1 2
20.19 odd 2 2800.2.g.r.449.4 4
35.3 even 12 1960.2.q.r.961.1 4
35.13 even 4 1960.2.a.s.1.2 2
35.18 odd 12 1960.2.q.t.961.2 4
35.23 odd 12 1960.2.q.t.361.2 4
35.27 even 4 9800.2.a.bu.1.1 2
35.33 even 12 1960.2.q.r.361.1 4
40.3 even 4 2240.2.a.bg.1.1 2
40.13 odd 4 2240.2.a.bk.1.2 2
60.23 odd 4 5040.2.a.by.1.1 2
140.83 odd 4 3920.2.a.bt.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
280.2.a.c.1.1 2 5.3 odd 4
560.2.a.h.1.2 2 20.3 even 4
1400.2.a.r.1.2 2 5.2 odd 4
1400.2.g.i.449.1 4 5.4 even 2 inner
1400.2.g.i.449.4 4 1.1 even 1 trivial
1960.2.a.s.1.2 2 35.13 even 4
1960.2.q.r.361.1 4 35.33 even 12
1960.2.q.r.961.1 4 35.3 even 12
1960.2.q.t.361.2 4 35.23 odd 12
1960.2.q.t.961.2 4 35.18 odd 12
2240.2.a.bg.1.1 2 40.3 even 4
2240.2.a.bk.1.2 2 40.13 odd 4
2520.2.a.x.1.2 2 15.8 even 4
2800.2.a.bk.1.1 2 20.7 even 4
2800.2.g.r.449.1 4 4.3 odd 2
2800.2.g.r.449.4 4 20.19 odd 2
3920.2.a.bt.1.1 2 140.83 odd 4
5040.2.a.by.1.1 2 60.23 odd 4
9800.2.a.bu.1.1 2 35.27 even 4